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Reconstructing the Girih Pattern (2): Underlying Polygonal Tiling

Reconstructing the Girih Pattern (2): Underlying Polygonal Tiling
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2019-03-06 20:35
PolygonVectorSolid GeometryTransformationGeometric Construction
① This example demonstrates how to reconstruct an Islamic girih pattern in GeoGebra, focusing on the underlying tiling composed of various polygons such as rectangles, parallelograms, and triangles. ② Core mathematical concepts include polygonal tiling, symmetry, congruence, and the use of 3D vectors for planar positioning, highlighting geometric construction and spatial relationships. ③ Construction details: Based on the summary, commands like Polygon, PolyLine, and Segment are used, creating multiple polygon objects (veelhoek1, veelhoek2, veelhoek3) and numerous points (A through S) and segments (f through s). The pattern is built by precisely placing vertices and connecting segments to form repeating units, with vectors or 3D elements aiding alignment. ④ Suitable for high school or university geometry courses, this example is ideal for exploring cultural mathematics, tiling art, and geometric construction, and can serve as material for interdisciplinary projects linking math and art.

GeoGebra Commands & Parameters

  1. Polygon(C, D, 5) → veelhoek1, f, g, h, i, j, E, F, G
  2. Polygon(H, I, 5) → veelhoek2, k, l, m, n, p, J, K, L
  3. Polygon(I, N, O, P, Q, R, S, M) → veelhoek3, i_1, n_1, o, p_1, q, r, s, m_1
  4. Polygon(Q, T, 5) → veelhoek4, t, a, b, c, d, U, V, W
  5. Polygon(U, Z, 5) → veelhoek5, e, f_1, g_1, h_1, j_1, A_1, B_1, C_1
  6. Polygon(D_1, E_1, 5) → veelhoek6, k_1, l_1, q_1, r_1, s_1, F_1, G_1, H_1
  7. Polygon(G_1, I_1, 5) → veelhoek7, t_1, a_1, b_1, c_1, d_1, J_1, K_1, L_1
  8. Polygon(M_1, N_1, 5) → veelhoek8, e_1, f_2, g_2, h_2, i_2, O_1, P_1, Q_1
  9. Polygon(M_1, R_1, 5) → veelhoek9, j_2, k_2, l_2, m_2, n_2, S_1, T_1, U_1
  10. Polygon(H, D, 5) → veelhoek10, p_2, q_2, r_2, s_2, t_2, V_1, W_1, Z_1
  11. Polygon(R_1, A_2, 5) → veelhoek11, a_2, b_2, c_2, d_2, e_2, B_2, C_2, D_2
  12. Polygon(C, E_2, F_2, G_2, D_1, H_2, V_1, I_2) → veelhoek12, c_3, f_3, g_3, h_3, i_3, j_3, v_1, k_3
  13. Polygon(S, W, 5) → veelhoek13, l_3, m_3, n_3, p_3, q_3, J_2, K_2, L_2
  14. Polygon(B_1, M_2, 5) → veelhoek14, r_3, s_3, t_3, a_3, b_3, N_2, O_2, P_2
  15. Polygon(Q_2, J_1, 5) → veelhoek15, d_3, e_3, f_4, g_4, h_4, R_2, S_2, T_2
  16. Polygon(E_1, F_2, 5) → veelhoek16, i_4, j_4, k_4, l_4, m_4, U_2, V_2, W_2
  17. Polygon(K_1, Z_2, Q_2, A_3, N_1, B_3, U_1, C_3) → veelhoek17, n_4, z_2, p_4, q_4, r_4, s_4, u_1, t_4
  18. Polygon(A_1, D_3, S_1, E_3, D_2, F_3, M_2, G_3) → veelhoek18, a_4, b_4, c_4, d_4, e_4, f_5, g_5, h_5
  19. PolyLine(H_3, L_3, D_4, N_3, M_3, C_4) → i_5
  20. PolyLine(I_3, C_4, K_3, A_4, B_4, J_3) → j_5
  21. PolyLine(W_3, V_3, R_3, T_3, U_3, Z_3) → k_5
  22. PolyLine(O_3, E_4, Q_3, S_3, R_3, P_3) → l_5
  23. Segment(H_3, C_4) → m_5
  24. Segment(C_4, I_3) → n_5
  25. Segment(I_3, J_3) → p_5
  26. Segment(N_3, O_3) → q_5
  27. Segment(O_3, P_3) → r_5
  28. Segment(S_3, F_4) → s_5
  29. Segment(T_3, G_4) → t_5
  30. Segment(W_3, Z_3) → a_5
  31. Segment(W_3, A_4) → b_5
  32. Segment(H_3, H_4) → c_5
  33. Segment(I_3, I_4) → d_5

AI Drawing Prompt Reference

Draw an Islamic geometric pattern composed of multiple regular polygons (including rectangles, parallelograms, and triangles). Start by drawing several pentagons, then connect the vertices to form repeating units. Use vectors to align the polygons, ensuring symmetry and consistency. Finally, construct the overall structure using polygons and segments.
The prompt above was compiled from the original drawing commands and is for reference only.